Clipboards in x11 are based on selections. There are two selections of note, the primary selection and clipboard selection. In GUI applications, selecting text copies it to the primary selection, middle clicking pastes the primary selection. Ctrl-C copies to the clipboard selection, Ctrl-V pastes the clipboard selection. The happy path is to stick to those operations only.
For terminals, ctrl-c/Ctrl-v have conflicting interpretations so are usually bound to ctrl-shift-c/ctrl-shift-v. Some (e.g. iterm) try to emulate copy on select on operating systems that don't support it by copying the selection to the single clipboard.
For vim, the selections are bound to buffers. * Is the primary/selection buffer, + is the clipboard. Use "+y/"+p for ctrl-c/ctrl-v equivalents, or "*y/"*p for copy-on-select/middle click equivalents. Note that some distros ship a default "vim-minimal"/"vim-tiny" package built without this support so they can use it as "vi" for POSIX compatibility without pulling in x11 libraries. In that case look for a "vim-x11", "vim-full" or "gvim" package. Or use neovim, I haven't seen packagers slice that up the same way. Depending on your terminal, you may be also able to use the GUI equivalents in insert mode anyway (don't try use your terminal paste in normal mode, it'll paste into the command buffer).
Shift-Insert is unfortunately application dependent. I think it _should_ be equivalent to middle click, but some apps treat it as ctrl-v.
Some clipboard managers or applications will overwrite the primary contents with the clipboard contents when you perform an explicit copy.
For terminals, ctrl-c/Ctrl-v have conflicting interpretations so are usually bound to ctrl-shift-c/ctrl-shift-v. Some (e.g. iterm) try to emulate copy on select on operating systems that don't support it by copying the selection to the single clipboard.
For vim, the selections are bound to buffers. * Is the primary/selection buffer, + is the clipboard. Use "+y/"+p for ctrl-c/ctrl-v equivalents, or "*y/"*p for copy-on-select/middle click equivalents. Note that some distros ship a default "vim-minimal"/"vim-tiny" package built without this support so they can use it as "vi" for POSIX compatibility without pulling in x11 libraries. In that case look for a "vim-x11", "vim-full" or "gvim" package. Or use neovim, I haven't seen packagers slice that up the same way. Depending on your terminal, you may be also able to use the GUI equivalents in insert mode anyway (don't try use your terminal paste in normal mode, it'll paste into the command buffer).
Shift-Insert is unfortunately application dependent. I think it _should_ be equivalent to middle click, but some apps treat it as ctrl-v.
Some clipboard managers or applications will overwrite the primary contents with the clipboard contents when you perform an explicit copy.